Cohomologie à support compact d’un espace au-dessus de l’immeuble de Bruhat-Tits de ${\mathrm{GL}}_n$ sur un corps local. Représentations cuspidales de niveau zéro.

Rajhi, Anis

Confluentes Mathematici, Tome 10 (2018) no. 1, pp. 95-124

Soit $\mathrm{G}$ le groupe ${\mathrm{GL}}_n\left(\mathrm{F}\right)$, où $\mathrm{F}$ est un corps localement compact non-archimédien, et $𝔅\left(\mathrm{G}\right)$ son immeuble de Bruhat-Tits. Nous construisons un complexe simplicial $\widetilde{𝒲}$, doté d’une action de $\mathrm{G}$ et d’une projection propre simpliciale $\mathrm{G}$-équivariante $p:\widetilde{𝒲}⟶𝔅\left(\mathrm{G}\right)$. Nous démontrons qu’en dimension supérieure la cohomologie à support compact ${\mathrm{H}}_c^{n-1}(\widetilde{𝒲},ℂ)$ contient comme sous-quotient toutes les représentations cuspidales irréductibles de niveau zéro.

Let $\mathrm{G}$ the group ${\mathrm{GL}}_n\left(\mathrm{F}\right)$, where $\mathrm{F}$ is a non-archimedean locally compact field, and $𝔅\left(\mathrm{G}\right)$ its Bruhat-Tits building. We construct a simplicial complex $\widetilde{𝒲}$, equipped with an action of $\mathrm{G}$ and with a $\mathrm{G}$-equivariant proper simplicial projection $p:\widetilde{𝒲}⟶𝔅\left(\mathrm{G}\right)$. We prove that the cohomology with compact support in higher dimensions ${\mathrm{H}}_c^{n-1}(\widetilde{𝒲},ℂ)$ contains as subquotients all irreducible cuspidal level zero representations.

Classification : 22E50
Mots-clés : Representations of the general linear $p$-adic groups, Bruhat-Tits buildings, Cohomology with compact support.

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